The well known geodesic structures pioneered by Buckminister Fuller, which comprises a single or double layers of geodesic structures including triangular substructures, provides two unique advantages which are, firstly, it enables one to construct a large scale unsupported dome without employing a long span beams or girders and, secondly, it provides a structure of highly futuristic appearance. While the conventional geodesic structure provides a unique way to built a large scale unsupported dome structure by using structural members of relatively small dimensions, it is less than an optimum structure in terms of the structural mechanics and less than ideal in terms of building economics. The geodesically reinforced honeycomb structure of the present invention has the same futuristic appearance as the conventional geodesic dome, while it provides a much improved strength to weight ratio and much better economics compared with the conventional geodesic structure.